Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 23 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

a day ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

December 22, 2019 at 12:07am

#971846

That's Funny - You Don't Look Druish

Normally, I pick things kinda randomly from my list of potential subjects, but as the solstice was at just 11:20 pm yesterday - not that long ago; in fact, it hasn't even happened yet as I start writing this - I wanted to do something appropriate for the season. The article is a couple of years old now, and then some, but it's still timely.

https://narratively.com/meet-the-modern-day-pagans-who-celebrate-the-ancient-god...

Meet the Modern-Day Pagans Who Celebrate the Ancient Gods

I mean, really, I thought pretty much everybody knew about this sort of thing by now, but apparently not.

First thing you see at that link is a photo. They're wearing nametags. For some reason I find that hilarious.

It is high summer, and we are at White Mountain Druid Sanctuary in southern Washington State. Under the immensity of the mountain, a couple of ramshackle barns stick up from the hayfields. Our priest, a straight-backed, snow-haired man, is delivering a homily on the attributes of the thunder god. Taranis, a powerful thunderbolt-tossing deity, is being honored at today’s solstice celebration because of his association with light, weather and sky.

To be fair, I hadn't heard of Taranis, and I've spent a good bit of time learning about different deities of different cultures. As with food, I suppose, there's always something else to learn. I like that.

There is a large wooden lodge with bed-and-breakfast facilities, meditation huts, and a stone circle straight out of Stonehenge, where, upon my arrival, about fifty people were pouring whiskey into deep wells and speaking Gaelic.

I understand the "sacrifice" aspect of modern Druidry, but I still gotta hope it was cheap whiskey.

The Druids I've known were obsessed with mead. Fine individuals, all of them, but as a dedicated aficionado of all things fermented and distilled, that was some of the worst mead I've ever had the dubious pleasure of sampling. I hope they didn't sacrifice that to the gods. I mean, if I were a god and had to drink that mead, I'd be tossing lightning bolts willy-nilly.

Loosely overseen by a central office – set in a back room in Thomas’ old house in Santa Fe, New Mexico – Ár nDraíocht Féin (ADF) is a polytheistic neo-pagan religion that draws its inspiration from ancient Indo-European traditions. It’s organized into local groups, called groves, and was founded in 1983 by a charismatic man named Isaac Bonewits, who, after completing a self-study program at UC Berkeley, earned a bachelor’s degree in – yes, really – Magic and Thaumaturgy.

True fact: I've met Isaac Bonewits. "Charismatic" is a decent adjective, but I have to raise a bit of an objection; that word is often applied to certain religious figures that have done... questionable things. Like start cults. So the implication is misleading. Bonewits wasn't a cult leader. In fact, he developed the best - in my humble opinion - system for identifying a religious group as a cult rather than a simple religious movement.

If you're interested, the framework he developed is here

.

This doesn't mean there aren't issues with Bonewits. I just wanted to dissociate what he did from any implication of "cult."

More and more in America, religion is something people choose (or don’t), rather than inherit. According to a 2015 study by the Pew Research Center, “As the Millennial generation enters adulthood, its members display much lower levels of religious affiliation, including less connection with Christian churches, than older generations.” However, the report also finds that many millennials remain spiritual in a broad sense, expressing “wonder at the universe” and an overall feeling of “gratitude” and “well-being.”

Here we go with the "generations" crap again. But okay, fine, I'll allow that there are differences between younger people and older people. Always have been. It's kind of our thing as humans.

It hits me that I am standing with a bunch of people I don’t know in the middle of a dark and remote farm being asked to drink unmarked liquid by a dude in a long white robe. The Water of Life shakes between my fingers.

Okay, I don't want to minimize the fears that people, perhaps especially women (like the author is, apparently), have in strange situations, but knowing Druids, she was probably safer in that circle than she would have been at, say, the supermarket.

Also, what is religion if not a way to address one's fears?

The diverse pantheon doesn’t phase anyone.

O Editor, where art thou? (It's "faze," not "phase." And yet she probably got paid for that article, while my lifetime sum total of income from writing is exactly 0, unless you count GPs.)

Still, there is tribalism in Druidry. Many of the practitioners I spoke with had the awkward, sharp, smart humor of the nerdy kids in middle school, which they wielded at me like little pikes, prodding and jabbing to see if I would laugh. Dr. Magliocco says this is partially constructed as a part of pagan identity. “Humor is a way that we mark insiders and outsiders,” she says. “A joke is a spell. Jokes clearly mark the boundaries. We can all laugh because we’re unusual, but we also draw a firm circle of who we are.”

Between all the drinking and joking and general nerdery, you'd think I'd make a perfect Druid. And yet I never quite fit in with the local Druids. I was more comfortable around Wiccans.

Another true story, then, concerning some local Druids. The grove I knew was based in Lynchburg, VA, about an hour from where I live. You may have heard of Lynchburg: it's the home of Liberty University, founded by professional asshole Jerry Falwell, and Falwell's Liberty Baptist Church, both of which are now headed by his son if I recall correctly. In other words, it's a major Southern Baptist center and the last place you'd expect to find Pagans, and yet, there they are.

I don't know how many states do this, but in VA, they let different groups "adopt" stretches of road, and the groups go out, like, two or four times a year and just pick up litter and whatnot to keep the road clean. It saves VDOT (and thus, taxpayers like me) money and brings the groups some exposure and the feeling that they're doing a good thing, so it's a win-win all around. Being a government run thing, they let pretty much anyone adopt the road, with certain exceptions for known hate groups and whatnot. So one time, a few years ago, these Druids noticed that the road past Liberty Baptist Church was up for adoption. And they started cleaning it up.

I still have the yellowed newspaper article discussing this magneted to my fridge, because to this day that situation gives me the warm fuzzies and cracks me up.

Anyone who's been following along with me here knows that I'm not a religious person. I find my meaning in the cycles of nature, in the vastness of the cosmos, in the rabid curiosity of science, and sometimes even in booze. But I also have a steadfast belief in freedom of religion: that idea, woven into the fabric of my country, that a person should be free to worship - or not - as one's heart dictates.

The solstice has passed now, and we are, as they put it in what has become my favorite Doctor Who special, "halfway out of the dark." To mark this occasion, I've been drinking beer. Quelle surprise, right? Point is, if I had a point when I started writing this, I've gotten sidetracked. We've completed another turn around the Sun. The Darkness is conquered, the Light once more prevails. It doesn't matter what you call yourself - Pagan, Christian, Muslim, Ba'hai, atheist, Hindu, animist, Druid, Jew, pantheist... the list goes on... what matters is that we're still all in this together, and we've done it. Another cycle.

Halfway out of the dark.

Well done. Well done.

Come on. Where else are you going to start with Druids and end with Doctor Who? Here. Only here.

Vale Sol Invictus!

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